To confirm the validity of the analytic continuation method, we re-examine the simulation done in Section V A of Ref. [1]} for the relaxation process of the magnetization of a \(N\)  spin-chain system. The Hamiltonian of the spin chain is
\(H_{\rm S} = - B_z \sum _{i=1}^N \sigma _i^z - \eta \sum _{i=1}^{N-1} \mathbf {\sigma }_i \cdot \mathbf {\sigma }_{i+1},\) 
